A Make-to-Order Capacitated Lot-Sizing Model with Parallel Machines, Eligibility Constraints, Extra Shifts, and Backorders

This study addresses the multi-period, multi-item, single-stage capacitated lot sizing problem (CLSP) in a parallel machine environment with machine eligibility constraints under a make-to-order production policy. A mixed-integer linear programming (MILP) model is developed to minimize total operational costs, including production, overtime, extra shifts, inventory holding, and backorders. The make-to-order setting introduces additional complexity by requiring individualized customer orders, each with specific due dates and product combinations, to be scheduled under constrained capacity and setup requirements. The model’s performance is evaluated in the context of a real-world production planning problem faced by a manufacturer of cold-formed steel profiles. In this setting, parallel forming machines process galvanized sheets of cold-rolled steel into a variety of profiles. The MILP model is solved using open-source optimization tools, specifically the HiGHS solver. The results show that optimal solutions can be obtained within reasonable computational times. For more computationally demanding instances, a runtime limit of 300 s is shown to improve solution quality while maintaining efficiency. These findings confirm the viability and cost-effectiveness of free software for solving complex industrial scheduling problems. Moreover, experimental comparisons reveal that solution times and performance can be further improved by using commercial solvers such as CPLEX, highlighting the potential trade-off between cost and computational performance.